## Problem of the Day #203: The Running RaceOctober 8, 2011

Posted by Saketh in : potd , trackback

Sid and Dasith are racing from point $A$ to point $B$, $1$ unit apart. Dasith gets to design the race track by picking any continous curve between $A$ and $B$, while Sid is allowed to take a “shortcut” once during the race.

A shortcut is any line segment parallel to $AB$ that starts and ends on the racetrack. Sid must announce the length $a<1$ of the shortcut he is going to take before Dasith designs the track.

Find all $a$ for which Sid will always find a place to use his shortcut.